In a bathroom, the biggest cause of a clogged drain is hair. Over-the-counter drain cleaners may solve the issue if you don't want to take the sink apart, but these usually are only a temporary fix. Here's what you would do to fix a clogged drain for a good while.
NOTE: If you have recently used a drain cleaner to no avail, take caution before taking the drain apart. If you get any of the chemical on your hands, immediately go to another sink and wash your hands. You may want to consider gloves to ensure you do not suffer any chemical burns.
The first step is to remove the trap, which is the U-shaped PVC tube under the sink. Place a bucket underneath the trap to catch any water caught in it. Remove what will most likely be two large nuts, and then remove the bottom part of the trap.
NOTE: Why is there a drain trap? Traps serve two very important purposes - prevention of both odors from the sewage system (whether you're hooked up to septic or city) from seeping into your bathroom, and insects (spiders mostly) from crawling in.
If its a slow drain (not completely clogged, but water takes longer than usual to clear the sink), its probably that the trap isn't clear.
The second place there could be a clog is near the stopper. The stopper is what you pull on the faucet to prevent water from leaving the sink. Remove it by loosening the packing nut securing the rod into the drain. There is a hole at the bottom of the stopper, and when the packing nut is loosened, the rod will slide out, allowing the stopper to be lifted out of the drain. Remove the gunk, put everything back together. Then test to see if there are any leaks by running the water.
NOTE: If you see any signs of a leak after you attach the drain trap, use a product like Pro-Dope to assist with sealing the threaded connections between the trap and the other PVC pipe pieces.